Wiener's test for random walks with mean zero and finite variance

Uchiyama, Kôhei

doi:10.1214/aop/1022855424

Cited by 10 publications

(10 citation statements)

References 5 publications

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“…This implies that the term in (19) is bounded by a constant multiple of exp{−c x/N 2 }, and is therefore also negligible compared to (21). This completes the bound for A j,k .…”

Section: Proof Of Propositions 16 and 17supporting

confidence: 56%

Moderate deviations for a random walk in random scenery

Fleischmann

Mörters

Wachtel

2008

Stochastic Processes and their Applications

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We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér's condition. We prove moderate deviation principles in dimensions d ≥ 2, covering all those regimes where rate and speed do not depend on the actual distribution of the scenery. For the case d ≥ 4 we even obtain precise asymptotics for the probability of a moderate deviation, extending a classical central limit theorem of Kesten and Spitzer. For d ≥ 3, an important ingredient in the proofs are new concentration inequalities for selfintersection local times of random walks, which are of independent interest, whilst for d = 2 we use a recent moderate deviation result for self-intersection local times, which is due to Bass, Chen and Rosen.

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“…This implies that the term in (19) is bounded by a constant multiple of exp{−c x/N 2 }, and is therefore also negligible compared to (21). This completes the bound for A j,k .…”

Section: Proof Of Propositions 16 and 17supporting

confidence: 56%

Moderate deviations for a random walk in random scenery

Fleischmann

Mörters

Wachtel

2008

Stochastic Processes and their Applications

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“…When

F

is arithmetic, the analogous result — that is, (10) holds if

h

is a multiple of the span of

F

— is verified in [22, Appendix B] with a relatively simpler proof. Remark Condition (Ha) implies

false({Ha}^{'} false) \exists x_{0} > 0, A false(x false) is positive for x ⩾ x_{0} and s.v. as x \to + \infty .

In fact

\log false[A (x) / A (x_{0}) false] = \int_{x_{0}}^{x} ε_{A} false(t false) d t / t

with

ε_{A} false(t false) : = t K false(t false) / A false(t false)

so that

A

is a (normalized) s.v.…”

Section: Introduction and Resultsmentioning

confidence: 84%

“…If

F

is arithmetic of span 1 and recurrent, we have the potential function

a false(x false) = \sum_{n = 0}^{\infty} false(P [S_{n} = 0] - P [S_{n} = - x] false)

which, well defined for every integer

x

, plays a crucial role in potential theory of random walk [20, Section 28]. If such an

F

satisfies (Ha) (

E | X |

may be finite), then

a (x)

admits an asymptotic estimate analogous to (7) (see [22, Theorem 7]).…”

Section: Introduction and Resultsmentioning

confidence: 99%

A renewal theorem for relatively stable variables

Uchiyama

2020

Bull. London Math. Soc.

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Let F {dx} be a relatively stable (r.s.) probability distribution on the whole real line and Sn the random walk started at the origin with step distribution F. We obtain an exact asymptotic form of the Green measure U {x + dy} = ∞ n=0 P [Sn − x ∈ dy] as x → ∞ when Sn is transient and Sn → ∞ in probability. If F is concentrated on [0, ∞), it is r.s. if and only if (x) := x 0 F {(t, ∞)}dt is slowly varying (s.v.) at infinity; our result entails that if F is non-arithmetic and r.s., then limx→∞ (x)U {[x, x + h)} = h for each h > 0. This surpasses the known result that assumes the stronger condition that xF {(x, ∞)} is slowly varying. An obvious analogue also holds for arithmetic variables.

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“…It turns out that the criterion does not depend on a random walk in question. This is the subject of the extension of the Wiener's test, proved in [34], that we state below. This invariance principle is one of the main tools we use in our investigation of the recurrence properties of the positive octant in Z n for R.…”

Section: Strong Law Of Large Numbersmentioning

confidence: 95%

Strong law of large numbers on graphs and groups

Mosina¹,

Ushakov

2011

Groups – Complexity – Cryptology

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Abstract. We consider (graph-)group-valued random element ξ, discuss the properties of a mean-set E(ξ), and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for ξ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.

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Wiener's test for random walks with mean zero and finite variance

Cited by 10 publications

References 5 publications

Moderate deviations for a random walk in random scenery

Moderate deviations for a random walk in random scenery

A renewal theorem for relatively stable variables

Strong law of large numbers on graphs and groups

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